It is reasonable to assume that \(F_i(y)\)  can be written as such as sum as we assume \(f\)  and \(y\)  to be analytic, which implies that \(F\)  is smooth. This means that a sufficiently rich basis of logistic and RBFs will approximate each \(F_i\)  in such a manner. In the real Koopman learning problem we would also have a constant and weighted sum of the measurements to help approximate each \(F_i\) . Using these extra terms to approximate \(F_i\)  in our analysis makes the math more cumbersome, and so we represent \(F_i\)  without them. The class of representable vector fields, \(F\) s, is extremely broad as both logistic and RBFs are universal function approximators [1]}, [2]}.
